Question Video: Determining the Associativity and Commutativity of Subtraction over the Real Numbers Mathematics • 9th Grade

True or False: 26 − (15 − √7) = (26 − 15) − √7. Is subtraction associative or nonassociative in ℝ? True or False: √7 − 15 = 15 −√7. Is subtraction commutative or noncommutative in ℝ?

04:40

Video Transcript

True or False: 26 minus the difference between 15 and the square root of seven is equal to the difference between 26 and 15 minus the square root of seven. Is subtraction associative or nonassociative in the set of real numbers? True or False: The square root of seven minus 15 is equal to 15 minus the square root of seven. Is subtraction commutative or noncommutative in the set of real numbers?

In the first part of this question, we are asked to determine if an equation holds true. One way of doing this is to try and rewrite both sides of the equation to be in the same form. To do this, let’s start by distributing the negative over the terms inside the parentheses on the left-hand side of the equation. This gives us 26 minus 15 plus the square root of seven. On the right-hand side of the equation, we can evaluate the difference between the integers inside the parentheses to obtain 11 minus the square root of seven.

We can simplify the expression on the left-hand side of the proposed equation to get 11 plus the square root of seven. We can see that this is not the same as 11 minus the square root of seven since the left-hand side is greater than 11 and the right-hand side is less than 11. Hence, the two sides of the proposed equation are not equal. So the answer is false.

In this second part of this question, we want to determine whether subtraction is an associative or nonassociative operation over the set of real numbers. We can recall that for an operation to be associative, we must be able to evaluate the operation in any order. So, we must have that for any real numbers 𝑎, 𝑏, and 𝑐, 𝑎 minus the difference between 𝑏 and 𝑐 is equal to the difference between 𝑎 and 𝑏 minus 𝑐. However, we have shown in the first part of this question that this does not hold true for all real values. In particular, it does not hold true when 𝑎 is equal to 26, 𝑏 is equal to 15, and 𝑐 is equal to the square root of seven. Hence, we have shown that the subtraction operation is nonassociative over the set of real numbers.

In the third part of this question, we need to determine if two sides of a proposed equation are equal. We can see that the two expressions are the same two real numbers. However, the order of the numbers has been reversed in the subtraction. There are many ways of showing that the two sides of the equation are not equal. One way is to note that the square root of seven lies between two and three, since two squared is less than seven, but three squared is greater than seven. Therefore, the square root of seven minus 15 will be less than zero, and 15 minus the square root of seven will be greater than zero. Since the left-hand side of the proposed equation is negative and the right-hand side is positive, they cannot be equal. So the answer is false.

In the final part of this question, we want to determine if the subtraction operation is commutative or noncommutative over the set of real numbers. We can recall that for an operation to be commutative, we need to be able to reorder the elements in the operation. Therefore, for subtraction to be commutative over the set of real numbers, we need 𝑎 minus 𝑏 to be equal to 𝑏 minus 𝑎 for all real numbers 𝑎 and 𝑏. However, we have already shown that this is not true when 𝑎 is equal to the square root of seven and 𝑏 is equal to 15. Hence, we have shown that the subtraction operation is noncommutative over the set of real numbers.